\newproblem{lay:7_2_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 7.2.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	What is the largest possible value of the quadratic form $5x_1^2+8x_2^2$ if $\mathbf{x}=(x_1,x_2)$ and $\mathbf{x}^T\mathbf{x}=1$, that is,
	if $x_1^2+x_2^2=1$? Try some examples of $\mathbf{x}$.
}{
   % Solution
	The matrix associated to this quadratic form and its orthogonal diagonalization is
	\begin{center}
		$A=\begin{pmatrix}5 & 0 \\ 0 & 8 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
		   \begin{pmatrix} 5 & 0 \\ 0 & 8 \end{pmatrix}
			 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}^T$
	\end{center}
	The maximum value of the quadratic form in this constrained optimization problem is equal to the value of the maximum eigenvalue, in this case 8, that is achieved 
	for $\mathbf{x}=(0,1)$, the eigenvector associated to the maximum eigenvalue. We show below the value of the quadratic form for a few values of $\mathbf{x}$
	\begin{center}
		$\begin{array}{lcl}Q(1,0)&=&5\cdot 1^2+8\cdot 0^2=5 \\
		  Q(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})&=&5\cdot (\frac{1}{\sqrt{2}})^2+8\cdot (\frac{1}{\sqrt{2}})^2=\frac{5}{2}+\frac{8}{2}=\frac{13}{2} \\
		  Q(0,1)&=&5\cdot 0^2+8\cdot 1^2=8\\
		\end{array}$
	\end{center}
}
\useproblem{lay:7_2_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

